PHYSICAL REVIEW E 95, 052415 (2017) Compressive elasticity of polydisperse biopolymer gels Xinpeng Xu * and Samuel A. Safran Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel (Received 4 January 2017; published 25 May 2017) We theoretically predict the nonlinear elastic responses of polydisperse biopolymer gels to uniaxial compression. We analyze the competition between compressive stiffening due to polymer densification by out-going solvent flow and compressive softening due to continuous polymer buckling. We point out that the polydispersity in polymer lengths can result in an intrinsic, equilibrium mode of nonaffine compression: nonuniform strain but with uniform force distribution, which is found to be more energetically preferable than affine deformation. In this case, the gel softens significantly after the onset of polymer buckling at small compression, but as compression increases, densification-induced stiffening becomes important and a modulus plateau should be observed for a large range of strain. We also relate our results to recent compression experiments on collagen gels and fibrin gels. DOI: 10.1103/PhysRevE.95.052415 I. INTRODUCTION Biopolymer gels comprise solvent and crosslinked network of stiff or semiflexible filaments (e.g., actin, collagen, and fibrin) [1]. They are important constituents of both the cellular cytoskeleton and extracellular matrix of tissues [2]. Unlike most synthetic gels comprising flexible polymers, biopolymer gels often exhibit highly nonlinear elastic responses to applied tensile forces above some small strain [1,3,4] or compressive forces at small strain [5,6], e.g., fibrin gels stiffen at small shear strains above 10% [1,3] and soften for even very small com- pression [5,6]. The strain stiffening nonlinearity of biopolymer gels is of biological significance [1,3] since it impacts force transmission by cells in such gels and many studies have aimed to understand its physical origin [1,3]. However, relatively less attention has been paid to the elastic response of biopolymer gels under compression [5,6]. This nonlinear elastic softening response is also critical for the physiological function of animal cells and some tissues [1,4,7,8]. For example, the nonlinear (asymmetric) responses of biopolymer gels to both tension and compression can together significantly increase the range of force transmission and are hence critical for the long-range cell-cell and cell-matrix communication [4,9,10]. Moreover, understanding of the elastic response of biopolymer gels to compression is a key feature of the self-contraction dynamics of the cellular cytoskeleton at subcellular scale [8]. Biopolymer gels usually have open network structures of much larger pore size (1 μm) than that of synthetic gels (∼10 nm). Larger pore sizes that usually imply smaller solvent-network friction facilitate interstitial solvent flows induced by gel deformation [6,8,11]. Biopolymer gels are thus effectively compressible at long enough time scales (e.g., about ∼ η s L 2 /G 0 ξ 2 ∼ 1 ms for a hydrogel of shear modulus G 0 ∼ 100 Pa, L ∼ 10 μm for typical cell size and pore size ξ ∼ 1 μm with water viscosity η s ∼ 10 −3 Pa s) [6,8,11]. Therefore, at long times, the Poisson ratio ν (or compressibil- ity) of biopolymer gels is controlled by the osmotic pressure difference and not by the molecular incompressibility; ν is thus small and can even be negative in the limit of moderately * Corresponding author: xpxu2010@gmail.com (a) (b) , , FIG. 1. Schematic illustrations of a polydisperse biopolymer gel of zero Poisson ratio under uniaxial compression: reference state (a) and deformed state (b). ℓ c and ℓ ′ c denote the undeformed (contour) length and the deformed length of polymer segments between cross- linkers, respectively. ξ eff and ξ ′ eff denote the effective mesh size of the undeformed and deformed gel, respectively. σ g (<0 for compression) is the external stress applied on the gel and ǫ g (<0 for compression) is the global strain of the gel due to the externally applied force. long time. Based on this we model biopolymer gels in this work as elastic materials having a small or negative Poisson ratio, and for simplicity we treat only the simplest case of zero Poisson ratio, i.e., ν = 0. But note that this simplification will not affect the generality of our predictions because a nonzero (small or negative) Poisson ratio can affect (decrease or increase) only the degree of stiffening by a factor 1 − 2ν . In addition, we consider equilibrated systems with no internal residual stress. This is appropriate for gels with relatively high concentrations of crosslinkers (i.e., high connectivity) [12]. In this case, stresses propagate quickly enough to equilibrate within the entire gel and the length of each polymer segment between crosslinkers is equal to its equilibrium value [1]. In contrast, computational studies on biopolymer gels of connectivity close to marginal stability are being carried out in other groups [6], for the case of significant internal residual stresses; in those cases, gel compression is highly nonaffine due to local force relaxation of the initial, residual stresses. Recent compression experiments [5,6] show that both collagen and fibrin gels soften gradually at small external compression and then reach a modulus plateau for a large 2470-0045/2017/95(5)/052415(11) 052415-1 ©2017 American Physical Society